Contributions to the mathematical biophysics of the central nervous system

Various neural mechanisms are considered which deal with point to point correspondence between two sets of neural elements with a smaller number of conducting elements between them; the transmission of nerve impulses in a limited range of intensities; movement of the transmission of excitation along a contour; the reaction to the size of an object independent of its distance; and an interpretation of the effect of a warning stimulus and of stimulus intensity upon reaction time. For the latter cases a comparison of the theoretical equations is made with some of the available experimental data, and a general agrement is found.     

Contributions to the mathematical biophysics of the cardiovascular system

The reflection of pressure waves in a fluid enclosed within a tube with an elastic wall is studied for the case of a localized change in diameter of the tube. The concept of impedance is introduced. The relation of the reflection characteristics of the parts of the tube at either side of the change is derived on the basis of the continuity of pressure and mass flow at the site of the change. This relations is used to derive the expression for the ratio of the pressure oscillations measured in front of, and behind, the constriction in terms of the constants of the system. As a result, a method is indicated to locate the coarctation from measurements of the pressures in front of, and behind it.     

Further contributions to a probabilistic interpretation of the mathematical biophysics of the central nervous system

Following a previous paper, equations are derived for the most probable time of firing of an efferent neuron in terms of the intensityE of excitation of the afferent pathway, whenE is either constant or any given function of time. The equations are not differential equations, but in integral form. It is suggested that ε, correspondinglyj, represent the number of excitatory, correspondingly inhibitory, terminal bulbs excited within the period of latent addition at a given most probable time. The relation between the suggested theory and the old one, based on differential equations for ε andj is discussed.     

Contributions to the mathematical biophysics of visual aesthetics

An expression for the intensity of central excitation corresponding to the perception of an angle is derived. A general expression for the aesthetic value of certain types of polygonal patterns is derived there-from and compared with experimental values obtained by the rank order method. The theory is found to agree with the experiment within a limit of about 10%.     

Contributions to the mathematical biophysics of branched circulatory systems

A derivation is given of the reflection coefficient of pressure waves in a vessel whose end branches into many smaller vessles. This coefficient depends on the number of these smaller vessels and their sizes relative to the size of the main vessel. Estimations are made of the order of magnitude of the coefficient. Assuming the main vessel to be of the order of size of an artery, it is shown that the reflection coefficient has a value close to one for reflections at branchings into vessels of arteriolar size. It is pointed out that the result may support the idea that the standing waves in the arterial system are due to reflections at the site of the arterioles.     

Suggestions for a mathematical biology of some pathological phenomena in the central nervous system,with special reference to some epilepsies

A neural model which was published in a preceding paper (Bull. Math. Biophysics,33, 539–553, 1971) and which explains the appearance of spontaneous sporadic outbursts of hyperactivity in some individuals, is elaborated further, with special reference tosome epilepsies. It is concluded thatsome epilepsies may be of purely psychogenic nature and amenable to psychotherapy. In particular such epilepsies may be arrested by appropriate conditioning. A case of such arrest has actually been described in medical literature.     

A reinterpretation of the mathematical biophysics of the central nervous system in the light of neurophysiological findings

The fundamental equations for the interaction between neurons used in mathematical biophysics seem at first incompatible with the actual neurophysiological findings on the synaptic transmission. It is shown, however, that those equations may be readily interpreted in terms of accepted neurophysiological views. What has been termed “synapse” in mathematical biophysics must be regarded as a complicated network of internuncial neurons. It is shown that, under rather conditions, the number of those interneurons willstatistically vary with time according to the differential equation postulated for the excitatory and inhibitory factors. The latter are thus interpreted as the number of excitatory and inhibitory interneurons.     

A suggestion for another statistical interpretation of the fundamental equations of the mathematical biophysics of the central nervous system

In a preceding paper an interpretation of the ε andj factors has been given in terms of an average effect of a large number of interneurons. In the present paper, a different interpretation is given in terms of the probability of a sufficient number of afferents to fire within the period of latent addition of the efferent. From this interpretation it follows that the old equations for ε andj are only first linear approximations to more complicated equations, the nature of which is suggested by this interpretation.     

Suggestions for a mathematical biophysics of auditory perception with special reference to the theory of aesthetic ratings of combinations of musical tones

Generalising the previously studied neurobiophysical schemes, consisting of excitatory and inhibitory elements, a neural mechanism is discussed, which may be involved in the perception of combinations of musical tones. Equations, giving the total value of central excitation for a combination of any two tones are derived, available observations are discussed in the light of the suggested theory.     

Functional morphology of the enteric nervous system with special reference to large mammals.

This short review reports the latest insights into the structural organization of the enteric nervous system, with special emphasis on the intrinsic innervation of the intestinal tract of large omnivorous mammals such as the pig. Using various techniques, including lesion experiments, morphological and neurochemical features of distinct neuronal populations as well as the direction of the axonal processes within the different nerve networks could be revealed. Special attention was paid to the considerable species differences in this respect between large omnivorous animals and humans on the one hand and small laboratory animals on the other hand.     

Immunohistochemical analysis of the rat central nervous system during experimental allergic encephalomyelitis, with special reference to Ia-positive cells with dendritic morphology

The rat central nervous system (CNS) during experimental allergic encephalomyelitis (EAE) was analyzed immunohistochemically from the preclinical to recovery stage by using monoclonal antibodies specific for rat T lymphocyte subsets and Ia antigen. Through combination of the avidin-biotin technique and carefully selected fixative, cells with dendritic morphology (DC) and infiltrating mononuclear cells were clearly and intensely demonstrated in the CNS parenchyma during EAE. In normal and complete Freund's adjuvant (CFA)-injected controls, there were no inflammatory foci. Ia (OX3)-positive parenchymal cells were not detected, whereas W3/25 stained DC that were located mainly in the white matter and W3/13 stained axons. At the preclinical stage, 11 days after CNS/CFA sensitization, a few clusters of Ia+ DC were detected in some sections of the spinal cord. The number of Ia+ DC increased as clinical signs developed (P less than 0.001). In rats with a clinical score of 1 or 2, Ia+ DC were mainly located in the perivascular region and closely associated with infiltrating T lymphocytes. However, at moribund state (score 3), Ia+ DC were evenly distributed in gray and white matter on almost all sections of the spinal cord. In recovered rats, the numbers of inflammatory foci and Ia+ DC were less than those in clinical EAE rats (P less than 0.001). Rats without clinical signs throughout the course also contained a few clusters of Ia+ DC. Double immunofluorescent staining with OX3 and anti-glial fibrillary acidic protein (GFAP) antiserum demonstrated that Ia+ DC were negative for GFAP. Their morphology and distribution were similar to those of nucleoside diphosphatase-positive cells, suggesting that Ia+ DC are microglia. In contrast to DC, no astrocytes or endothelial cells express detectable levels of Ia antigen in control and clinical EAE rats. These findings suggest that brain cells other than Ia+ DC may not be involved in the local immune interaction. Ia+ DC may play a significant role in antigen presentation in the CNS with EAE.